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I was reading through Akhiezer's book Lectures on Integral Transforms and in chapter nine, he states that the Hankel transform is unitary for $\nu > -1$, so that for a suitable function, $f$,

$f(y) = \int_0^{\infty}dr \sqrt{ry}J_{\nu}(ry) \int_0^{\infty} dx f(x) \sqrt{xr} J_{\nu}(xr)$.

He leaves the proof that it is unitary to the reader, but I cannot for the life of me even begin to know what the angle of attack should be. I've done some pretty extensive Googling, but have come up completely short. Can anyone point me in the right direction or if you know where to find the proof itself, can link me to it? Much appreciated.

I have found a couple of proofs for the case where $\nu > -1/2$, but none for the range $-1 < \nu < -1/2$.

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2 Answers

up vote 1 down vote accepted


This ensures orthogonality. You should think in terms of linear algebra - transition matrix to any orthogonal basis in orthogonal. So you need to prove orhogonality of Bessels. This is stated in Wiki Link above.


Actually I do not quite understand where this orthogonality comes from ? Can one comment ?

Because usually orthogonality comes easily - the eigenfunctions of some symmetric matrix (operator) are orthogonal. But Bessels are eigenfucntions in $\nu$ , not in "x". So I am puzzled.

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That is basically what I want to prove, however the proofs I have seen all deal with the $\nu > -1/2$ and I'm not sure how to prove it for the more general case $\nu > -1$. The fact that it is true (at least according to Akhiezer and the NIST Handbook of Mathematical Functions and I checked it with Mathematica for some functions) is great, but I'm interested in the proof itself as I need to adapt it for research purposes. –  Cameron Williams Jan 10 '12 at 18:25
I see your point now. Hankel can be seen as FT for functions in R^3 depending on radius only. Would it be helpful ? What is the meaning of $\nu=-1/2$ ? –  Alexander Chervov Jan 11 '12 at 5:59
Sorry for the delayed response, but I didn't actually need to use the proof itself. I just needed to use the result. Apparently I had goofed when doing some Dirac delta algebra, but it all worked out. Thanks for your input anyway! –  Cameron Williams Mar 7 '12 at 0:37
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you can see the book:

Integral Transforms for Engineers By Larry C. Andrews, Bhimsen K.

best regard, Ramin

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Hi Ramim, it turned out that I did not need the proof after all (as I went a more measure-theoretic route). Thanks for the reference though. :) –  Cameron Williams Jan 14 at 16:44
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